In an interview, are you expected to populate data structures so you can test your code?

I've worked at the same company as a Web Developer for a VERY long time and am now looking around, so I haven't encountered many interview tests. I'm doing the test questions in Programming Interviews Exposed, but am curious if in the actual interview your expected populate data structures before manipulating them, etc.

I understand that it's different in each company and some questions will be answered on a white board, but in cases in which you're expected to test your code would they provide you with some code that populates the data structure?

For example, say you're asked to find the lowest common ancestor for 2 nodes in a BST. Would I be expected to first create a BST class and then populate it with data or would the interviewer provide me with that part of the puzzle?